Numerical Investigation of Convection Pseudospectral Methods

7/25/2019 Numerical Investigation of Convection Pseudospectral Methods

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Internal Heating

Spectral Chebyshev Integration

Application

Acknowledgments

Numerical Investigation of ConvectionApplication of Chebyshev Integration

B. Cloutier1

H. Johnson 4

B. Muite 4

P. Rigge2

J. Whitehead3

1Department of PhysicsUniversity of Michigan

2Department of Computer ScienceUniversity of Michigan

3Department of MathematicsUniversity of Michigan

4Department of Mathematics

University of Massachusetts
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Application

Acknowledgments

Overview

1 Internal Heating

Equations of Motion

2 Spectral Chebyshev IntegrationChebyshev Polynomials

Solving Vorticity Equation

Timestepping Scheme

3 Application

4 Acknowledgments
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Equations of Motion

Assumptions

Convection driven solely by internal heating infinite Pr

(similar to the Earths mantle)

Start with dimensionless two dimensional Navier-Stokes

equations with the Boussinesq approximation.

Choose time-scale of h2

/, length scale of handtemperature scale of Hh2/. Where his height, isthermal diffusivity, and H is heating.

2= RTx (1)

Tt+ z

Tx x

Tz=

Txx+

Tzz+1 (2)

|z=1,1=0 z|z=1,1 =0 (3)

T|z=1,1 =0 (4)

T(x, z)- Temperature. (x, z)- Streamfunction,

u(x, z) = (u,w)
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Equations of Motion

Biharmonic Equation

Decompose biharmonic into two equations, one for the

stream function and the other for the vorticity

2

=f

(x,z

)= (5)

= f(x, z) (6)

Easy to do with free slip boundary conditions

Four order problem with N nodes implies dividing by N4

N=4096, 1N4 3.5 1015, 1

N2 6 108 (edge of double

percision)
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Acknowledgments

Solving Vorticy Equation

Timestepping Scheme

Chebyshev polynomials

Consider a problem on the interval [1,1]

Space is discretized using Chebyshev polynomials

Tn(z) :=cosncos1 z (7)

with xevaluated at Chebyshev points

zi :=cos i

N

i=0, ...,N (8)

Discretization allows for the use of Fast Fourier Transform

to calculate integrals and derivatives

I l H i
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Timestepping Scheme

Fourier Space

FFT in the x-direction and rewriting derivatives 1

(ikx)2+ zz= f(x, z) (9)

(x, z)and f(x, z)are periodic functions on the interval[1,1].

1

Properties of FFT allow

nf(x)

xn = (ik)n

f(x)

I t l H ti
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Acknowledgments


Timestepping Scheme

Chebyshev Integration

Chebyshev integration matrix method amounts to solving

for the highest order derivative by expanding as a

summation of Chebyshev polynomials in z-direction.

((ik)2I2+ I0+ LBC)zz= f(x, z) + RBC (10)

LBCand RBCrepresent boundary conditions

zzis a vector of the truncated series expansion for zz.

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Timestepping Scheme

Finding Numerical Solution

Given Boundary conditions, fix two coefficients from the

indefinite integral ofzz.Linear system is solved to find zzand then integrated tofind .

Lastly we can use IFFT to convert back to real space and

find(x,z

).

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Timestepping Scheme

Finding I0and I2

Suppose, where bnare Chebyshev series expansion

coefficients for f

zz=n=1

bnTn(z) (11)

We use the following indefinite integral identities

(12)

T0(z) = T1(z), T1(z) =T2(z)

4 Tn(z) =

Tn+1(z)

2(n+1)

Tn1(z)

2(n 1)

We can use these integral identities in write integral

matricies ofzz,z, and

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Timestepping Scheme

Using Indefinite Integral Identities

Truncated to N+3 modes

Ux= e1+ (b0 b2

2 )T1(x) +

N+3n=2

(bn1 bn

+1

2n Tn(x)

U= e0+ (e1 b1

8 +

b3

8)T1(x) + (

b0

4

b2

6 +

b4

24)T2(x) + ...

+

N+3n=3

bn2

4n(n 1) bn

2(n 1)(n+1)+ b

n+24n(n+1)

Tn(x)

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Timestepping Scheme

Integration Matricies Explicitly

Resulting system of equations are

((ik)2I2+ I0+ LBC+ LBC)zzzz= f(x, z) + RBC (13)

(ik)2b0+ e0 = f0 (14)

(ik)2b1+ (e1 b1

8 +

b3

8) = f1 (15)

(ik)2b2+b

04

b2

6 + b

424

= f2 (16)

And for 2< n N, use formula were bn=0 for n> Nandfnare Cheby expansion coefficients

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Timestepping Scheme

Impose Boundary Conditions

All that is left is to impose the two boundary that will fix the

last 2 coefficientsUsing the following boundary conditions,(1)we have,

(1) = e0 (e1 b1

8+

b3

8) + (

b0

4

b2

6

b4

24+ ... (17)

+n=3

(1)n

bn2

4n(n 1)

bn

2(n 1)(n+ 1)+

bn+2

4n(n+ 1)

Tn(x) (18)

x(1) = e1 (b0 b2

2

) +

n=2

(1)n

(bn1 bn+1

2n

Tn(x) (19)

xx(1) =N

n=1

(1)nbn (20)

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Application

Acknowledgments


Timestepping Scheme

Timestepping: Implicit Midpoint Rule

Fixed point iteration we solve

Tn+1 Tn

dt +

1

2un+1 Tn+1 +

1

2un Tn =

1

2(Tn+1 + Tn)

2n+1 = f(x, z)n+1

n+1 = f(x, z)n+1

n+1 =n+1

Initial conditions: perturbation of conductive solution

T(x, z) =1

2(1 z2) + .1sin

2x

xmax

sin(.5(1 +z))

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Application

Acknowledgments

Scaling

Parrallel code run Trestles and Lonestar. Thanks toTeragrid/Xsede resources supported by award

TG-CTS1100010.

101

102

103

101

102

103

104

Cores

Runtime

N=10242

N=20482

N=40962

Ideal Scaling

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Acknowledgments

Aspect Ratio: 8, Rayleigh number:105, top is temperature field

and bottom is the streamfunction.

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Acknowledgments

Aspect Ratio: 4, Rayleigh number:105, top is temperature field

and bottom is the streamfunction.

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S l Ch b h I i
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Application

Acknowledgments

Video

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S t l Ch b h I t ti
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Acknowledgments

Acknowledgments

We would like to University of Michigan UndergraduateResearch Opportunities Program and APS.

This work was performed on Teragrid/Xsede resources

supported by award TG-CTS1100010

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Application

Acknowledgments

D.J. Tritton,

Physical Fluid Dynamics, Oxford, (1988).J.P. Whitehead and C.R. Doering,

Internal heating driven convection at infinite Prandtl

number, JMP, (2011).

L.N. Trefethen,Spectral Methods in Matlab, SIAM, (2000).

B.K Muite

A numerical comparison of Chebyshev methods for solving

fourth order semilinear initial boundary value problems,JCAM, (2009).
http://find/

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Numerical Investigation of Convection Pseudospectral Methods

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